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{
"Title": "Complete elliptic integral of the first kind $K(m_r)$ at the singular values",
"Definition": "For $r\\in\\mathbb Z_{>0}$, set $m_r=k_r^2=\\lambda(i\\sqrt r)$, where $\\lambda$ is the elliptic modular function CITE{DLMFModularFunctions}. The entry is $K(m_r)$, classically written $K(k_r)$.",
"Parameters": {
"r": {
"type": "Z",
"title": "index, with $\\tau=i\\sqrt r$",
"display": "$r$",
"constraints": "positive integer"
}
},
"Comments": {
"comment-convention": "This table writes $K(m_r)$ throughout: $K$ takes the parameter $m_r=k_r^2$, the convention of arb's `elliptic_k`, of HREF{Complete_elliptic_integral_of_the_first_kind_K}[the complete elliptic integral $K(m)$], and of the elliptic alpha function in HREF{T299}[its table]. The classical literature on singular values, and DLMF CITE{DLMFEllipticIntegrals}, write the same number $K(k_r)$ with the elliptic modulus $k_r$. The two notations name one number, but read in the wrong convention $K(k_r)$ is a different one: $K$ at parameter $k_r$.",
"comment-cm-point": "The lattice $\\mathbb Z+\\mathbb Z\\,i\\sqrt r$ has complex multiplication by the order $\\mathbb Z[\\sqrt{-r}]$, of discriminant $-4r$."
},
"Formulas": {
"formula-theta": "$K(m_r)=\\frac{\\pi}{2}\\theta_3(0,q)^2$, where $q=e^{-\\pi\\sqrt r}$ and $\\theta_3(0,q)=\\sum_{n\\in\\mathbb Z}q^{n^2}$ CITE{MathWorldSingularValue}.",
"formula-complement": "$K(1-m_r)=\\sqrt r\\,K(m_r)$.",
"formula-hypergeometric": "$K(m_r)=\\frac{\\pi}{2}\\,{}_2F_1(1/2,1/2;1;m_r)$ CITE{DLMFHypergeometric}."
},
"Programs": {
"program-sage": {
"language": "Sage",
"code": "import numberdb.sage as numberdb\nfrom sage.rings.complex_arb import ComplexBallField\n\nr = 2\nCBF = ComplexBallField(numberdb.bits(100, losing=256))\ntau = CBF.gen(0) * CBF(r).sqrt()\nm = tau.modular_lambda().real()\nCBF(m).elliptic_k().real()"
}
},
"Similar tables": [
{
"table": "HREF{T295}[Singular values $k_r$ of the elliptic modulus]",
"relation": "holds $k_r$ and $m_r=k_r^2$; the entries here are $K$ at those $m_r$"
},
{
"table": "HREF{Complete_elliptic_integral_of_the_first_kind_K}[Complete elliptic integral of the first kind $K(m)$]",
"relation": "that table stores the same function at rational parameters; the entry $m=1/2$ equals the $r=1$ entry here"
},
{
"table": "HREF{Values_of_Ramanujan's_signature_hypergeometric_functions}[Values of Ramanujan's signature hypergeometric functions $F_r(x)$]",
"relation": "signature $2$ satisfies $F_2(m)=2K(m)/\\pi$ by CITE{formula-hypergeometric}"
},
{
"table": "HREF{Values_of_the_Gamma_function_at_rational_numbers}[Values of the Gamma function at rational numbers]",
"relation": "the low-index singular values have Gamma-product closed forms"
},
{
"table": "HREF{Polya's_random_walk_constants}[Pólya's random walk constants]",
"relation": "the return probability is $p(3)=1-1/u(3)$, where Watson's integral is $u(3)=\\frac{12}{\\pi^2}(18+12\\sqrt2-10\\sqrt3-7\\sqrt6)K(m_6)^2$"
}
],
"Links": {
"DLMFEllipticIntegrals": {
"title": "DLMF 19.2: Definitions of complete elliptic integrals",
"url": "https://dlmf.nist.gov/19.2"
},
"DLMFHypergeometric": {
"title": "DLMF 19.5: Hypergeometric function representations",
"url": "https://dlmf.nist.gov/19.5"
},
"DLMFModularFunctions": {
"title": "DLMF 23.15: modular functions and the elliptic modular function $\\lambda$",
"url": "https://dlmf.nist.gov/23.15"
},
"MathWorldSingularValue": {
"title": "MathWorld: Elliptic Integral Singular Value",
"url": "https://mathworld.wolfram.com/EllipticIntegralSingularValue.html"
}
},
"Keywords": [
"elliptic integral singular value",
"Chowla-Selberg",
"theta function",
"lemniscate",
"complete elliptic integral singular value"
],
"Tags": [
"special values",
"modular function",
"elliptic curves",
"special functions"
],
"Data properties": {
"type": "R",
"rigour": "proven",
"complete": "no",
"complete-note": "it holds $K(m_r)$ for every integer $1\\leq r\\leq100$, matching HREF{T295}[the singular moduli $k_r$]; over this range the order $\\mathbb Z[\\sqrt{-r}]$ has class number at most $12$",
"rigour details": "Each value is computed in Sage's arb-backed `ComplexBallField` at `numberdb.bits(digits, losing=256)` bits. The generator evaluates $m_r=\\lambda(i\\sqrt r)$ as a complex ball, checks that the imaginary part contains zero, and evaluates arb's `elliptic_k` at $m_r$. Before the values were sent, the generator checked CITE{formula-theta}, CITE{formula-complement}, and CITE{formula-hypergeometric} on every computed row."
},
"Display properties": {
"number-header": "$K(m_r)$"
},
"Numbers": {
"1": {
"number": "1.854074677301371918433850347195260046217598823521766905585928045056021776838119978357271861650371897",
"comment": "$K(m_1)=\\Gamma(1/4)^2/(4\\sqrt\\pi)$ CITE{MathWorldSingularValue}.",
"equals": "HREF{Complete_elliptic_integral_of_the_first_kind_K#1/2}"
},
"2": {
"number": "1.645568395293458039866051685287072715999557026055401037265292137149578863729330871593184129832048067",
"comment": "$K(m_2)=\\sqrt{1+\\sqrt2}\\,\\Gamma(1/8)\\Gamma(3/8)/(2^{13/4}\\sqrt\\pi)$ CITE{MathWorldSingularValue}."
},
"3": {
"number": "1.598142002112540144460965105390112888900611097136742843395001850717051182376299048153673076745023371",
"comment": "$K(m_3)=3^{1/4}\\Gamma(1/3)^3/(2^{7/3}\\pi)$ CITE{MathWorldSingularValue}."
},
"4": {
"number": "1.582551727223715911833135071070409876529488149618789243497169448478208535186663551736209814065543222",
"comment": "$K(m_4)=(1+\\sqrt2)\\Gamma(1/4)^2/(2^{7/2}\\sqrt\\pi)$ CITE{MathWorldSingularValue}."
},
"5": {
"number": "1.576390394802592476323757555260863247741937034744074005958674897925807012418125362905820159225379507",
"comment": "$K(m_5)=(2+\\sqrt5)^{1/4}\\sqrt{\\Gamma(1/20)\\Gamma(3/20)\\Gamma(7/20)\\Gamma(9/20)/(160\\pi)}$ CITE{MathWorldSingularValue}."
},
"6": {
"number": "1.573656231264378266467658770764262615447771495128054395083197322369080422472509636815994782912924097",
"comment": "$K(m_6)=\\sqrt{(\\sqrt2-1)(\\sqrt3+\\sqrt2)(2+\\sqrt3)}\\sqrt{\\Gamma(1/24)\\Gamma(5/24)\\Gamma(7/24)\\Gamma(11/24)/(384\\pi)}$ CITE{MathWorldSingularValue}."
},
"7": {
"number": "1.572339753406137527416490497176982668311869155892281594080158998982190209092426217240368536444070701",
"comment": "$K(m_7)=\\Gamma(1/7)\\Gamma(2/7)\\Gamma(4/7)/(4\\cdot7^{1/4}\\pi)$ CITE{MathWorldSingularValue}."
},
"8": {
"number": "1.571665689206437042031861058437591105397806921254527952296192373178656438735452990723968471906921488",
"comment": "$K(m_8)=\\sqrt{(2\\sqrt2+\\sqrt{1+5\\sqrt2})/(4\\sqrt2)}(1+\\sqrt2)^{1/4}\\Gamma(1/8)\\Gamma(3/8)/(8\\sqrt\\pi)$ CITE{MathWorldSingularValue}."
},
"9": {
"number": "1.571303417736683352208481497338796809391174821642956915852621765840171701094284147643588839960207742",
"comment": "$K(m_9)=3^{1/4}\\sqrt{2+\\sqrt3}\\,\\Gamma(1/4)^2/(12\\sqrt\\pi)$ CITE{MathWorldSingularValue}."
},
"10": {
"number": "1.571100880616723881003002373510769622686969652665615951839918153838543584392595612006151433598168530",
"comment": "$K(m_{10})=\\sqrt{2+3\\sqrt2+\\sqrt5}\\,\\sqrt{\\Gamma(1/40)\\Gamma(7/40)\\Gamma(9/40)\\Gamma(11/40)\\Gamma(13/40)\\Gamma(19/40)\\Gamma(23/40)\\Gamma(37/40)/(2560\\pi^3)}$ CITE{MathWorldSingularValue}."
},
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}